Problem: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{6z^2 - 96}{z^3 + 13z^2 + 36z}$
First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {6(z^2 - 16)} {z(z^2 + 13z + 36)} $ $ p = \dfrac{6}{z} \cdot \dfrac{z^2 - 16}{z^2 + 13z + 36} $ Next factor the numerator and denominator. $ p = \dfrac{6}{z} \cdot \dfrac{(z + 4)(z - 4)}{(z + 4)(z + 9)}$ Assuming $z \neq -4$ , we can cancel the $z + 4$ $ p = \dfrac{6}{z} \cdot \dfrac{z - 4}{z + 9}$ Therefore: $ p = \dfrac{ 6(z - 4)}{ z(z + 9)}$, $z \neq -4$